Aimed at the problems that transmission data is liable to make mistakes or to lose caused by fading and interference in high-speed aviation communication scenario, a new alternating direction method of multipliers decoder for non-binary low-density parity-check codes in Galois fields F2q is proposed. Firstly, a bit embedding rule is proposed, and the procedure of formulating the maximum likelihood decoding problem to a linear integer problem in real space is presented. Secondly, after relaxing the integer problem to a continuous one, an efficient ADMM algorithm is customized to solve the latter, where all the entries of the variable vectors can be obtained in parallel. Thirdly, it is important that the proposed ADMM decoder satisfies the needs of favorable codeword-independent property under some mild conditions and its computation complexity in each ADMM iteration is roughly O(nq). The simulation results show that this ADMM decoder is prior to the others in error-correction and decoding efficiency.